A119304 Triangle read by rows: T(n,k) = binomial(4n-k,n-k), 0 <= k <= n.

1, 4, 1, 28, 7, 1, 220, 55, 10, 1, 1820, 455, 91, 13, 1, 15504, 3876, 816, 136, 16, 1, 134596, 33649, 7315, 1330, 190, 19, 1, 1184040, 296010, 65780, 12650, 2024, 253, 22, 1, 10518300, 2629575, 593775, 118755, 20475, 2925, 325, 25, 1, 94143280, 23535820

Offset: 0

Author

Paul Barry, May 13 2006

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins
       1;
       4,     1;
      28,     7,    1;
     220,    55,   10,    1;
    1820,   455,   91,   13,   1;
   15504,  3876,  816,  136,  16,  1;
  134596, 33649, 7315, 1330, 190, 19, 1;
		

Links

• Indranil Ghosh, Rows 0..125, flattened

Crossrefs

Rows sums are A052203. First column is A005810. Inverse of A119305.

Cf. A092392, A119301.

Programs

Mathematica

Flatten[Table[Binomial[4n-k,n-k],{n,0,9},{k,0,n}]] (* Indranil Ghosh, Feb 26 2017 *)

PARI

tabl(nn) = {for (n=0,nn,for (k=0,n,print1(binomial(4*n-k,n-k),", ");); print(););} \\ Indranil Ghosh, Feb 26 2017

Python

from sympy import binomial
i=0
for n in range(12):
    for k in range(n+1):
        print(str(i)+" "+str(binomial(4*n-k,n-k)))
        i+=1 # Indranil Ghosh, Feb 26 2017

Formula

Riordan array (1/(1-4f(x)),f(x)) where f(x)(1-f(x))^3 = x.

From Peter Bala, Jun 04 2024: (Start)

'Horizontal' recurrence equation: T(n, 0) = binomial(4*n,n) and for k >= 1, T(n, k) = Sum_{i = 1..n+1-k} i*(i+1)/2 * T(n-1, k-2+i).

T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(3*n-k-j, 2*n). (End)